Solve equation with Poisson Sum?

Questions
1

Consider the following code:

In[263]:= Clear["Global`*"]
            m = 2;
            a = 1/20;
            k = 0;
        
        Solve[Sum[PDF[PoissonDistribution[p], i], {i, k, m}] == a, p]
    
        During evaluation of In[263]:= Solve::nsmet: This system cannot be solved with the methods available to Solve.
        
    Out[267]= Solve[E^-p + E^-p p + 1/2 E^-p p^2 == 1/20, p]

I’ve tried many ways but haven’t succeed. So how to solve this equation?

2

Your function has negative derivative and it vanishes at +infinity. There is no real solution. You can find complex solutions if you give a bound for p:

Solve[Sum[PDF[PoissonDistribution[p], i], {i, k, m}] == a && 
  Abs[p] < 2, p]
3

Internet version of mathematica finds the solution p=6.29 as it should be. But it writes that this solution is over reals, but 6.29 is real number, not comlex. So my question is how to find this 6.29 solution?

4

Sorry, I had misread the equation. Simply enlarge the interval:

Solve[Sum[PDF[PoissonDistribution[p], i], {i, k, m}] == a && 
  0 < p < 7, p]